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scheduling in wireless sensor networks

Olesia Mokrenko, Carolina Albea-Sanchez, Suzanne Lesecq, Luca Zaccarian

To cite this version:

Olesia Mokrenko, Carolina Albea-Sanchez, Suzanne Lesecq, Luca Zaccarian. A hybrid control law

for energy-oriented tasks scheduling in wireless sensor networks.

IEEE Transactions on Control

Systems Technology, Institute of Electrical and Electronics Engineers, 2018, 26 (6), pp.1995-2007.

�10.1109/TCST.2017.2750999�. �hal-01402876�

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A hybrid control law for energy-oriented tasks

scheduling in wireless sensor networks

Olesia Mokrenko, Carolina Albea, Suzanne Lesecq, Luca Zaccarian

Abstract—Energy is a key resource in Wireless Sensor Net-works (WSNs), especially when sensor nodes are powered by batteries. This work investigates how to save energy of the whole WSN, thanks to control strategies, in real time and in a dynamic way. The energy management strategy is based on a Hybrid Dynamical System (HDS) approach. This choice is motivated by the hybrid inherent nature of the WSN system when energy management is considered. The hybrid nature basically comes from the combination of continuous physical processes, namely, the charge/discharge of the node batteries; while the discrete part is related to the change in the functioning modes and an Unreachable condition of the nodes. This approach provides a decentralized controller with low computational load that reduces the number of switching as compared to existing approaches. The proposed strategy is evaluated and compared in simulation on a realistic test-case. Lastly, they have been implemented on a real test-bench and the obtained results have been discussed.

Index Terms—Hybrid Dynamical Systems, Wireless Sensor Network, Energy management

I. INTRODUCTION

Technological and technical developments performed in the areas of wireless communication, micro-electronics and system integration have led to the advent of a new generation of large-scale sensor networks suitable for various applica-tions [1]. A set of small electronic devices (the so-called sensor nodes), autonomous, equipped with sensors and able to communicate with each other wirelessly forms a Wireless Sensor Network (WSN) capable of monitoring a phenomenon of interest, and possibly react to the environment. They can provide high level information about this phenomenon to users by the combination of measurements taken by the various sensors and, then, communicate via the wireless medium. This technology promises to revolutionize our way of life, working and interacting with the physical environment around us. Sensor nodes able to communicate wirelessly, together with distributed computing capabilities, allow developing new applications that were impractical or too expensive a few years ago. Today, tiny and inexpensive sensor nodes can be literally scattered on roads, bridges, buildings wings of planes or forests, creating a kind of “second digital skin” that can

Olesia Mokrenko was with Univ. Grenoble Alpes, F-38000 Grenoble, France / CEA, LETI, Minatec Campus while this work was performed.

Carolina Albea is with LAAS-CNRS, Universit´e de Toulouse, CNRS, UPS, Toulouse, France

Suzanne Lesecq is with Univ. Grenoble Alpes, F-38000 Grenoble, France / CEA, LETI, Minatec Campus.

Luca Zaccarian is with LAAS-CNRS, Universit´e de Toulouse, CNRS, UPS, Toulouse, France and also with the Dipartimento di Ingegneria Industriale, University of Trento, Italy

Manuscript received July XX, 2016; revised DD MM, YYYY.

detect various physical phenomena such as vibrations created by earthquakes or the change in the shape of a mechanical structure, or fire appearance and evolution in forests. As a consequence, many applications deal with the detection and monitoring of disasters (earthquakes, floods), environmental monitoring and mapping of biodiversity, intelligent build-ings, advanced farming techniques, surveillance and preventive maintenance of machinery, medicine and health, logistics and intelligent transportation systems.

WSNs are often characterized by a dense deployment of nodes in large-scale environments with various limitations. These limitations are related to processing and memory ca-pabilities, radio communication ranges, but also to energy resources as the sensor nodes may be powered by batteries. Note that even if sensor nodes are connected to power lines, they must be power-efficient because it is not acceptable from an ecology viewpoint, to drastically increase the number of power plants just to feed all these new devices. For sensor networks powered by batteries, changing the batteries has an extra cost related to their recharge and/or replacement that must be taken into account. Moreover, as the batteries in the network will certainly not be drained at the same time, the network maintenance teams change all the batteries during a unique intervention, leading to a suboptimal discharge of some of the batteries. Also, the sensor nodes may be placed in locations that are hard to access, for safety or economic reasons. Indeed, it is widely recognized that the energy limitation is an unavoidable issue in the design and deployment of WSNs because it imposes strict constraints on the network operation. Basically, the power consumption of the sensor nodes plays an important role in the life of the network. This aspect has become the predominant performance criterion for sensor networks. If we want the sensor network to perform its functionality satisfactorily as long as possible, these energy constraints imply trade-offs among different activities both at the node level and at the network layers.

The WSN lifespan increase has already been addressed in the literature, from sensor-level to network-level [2]–[4]. [5] provides an overview of these techniques. We employed Model Predictive Control (MPC) [6] in a WSN composed of three states: 2 functioning modes (Active and Standby) and an Unreachable condition due to environmental disturbances as communication breakdown, or an insufficient energy level [7]. This MPC controller selects the devices to be in Active mode both to limit the WSN overall energy consumption and extend its lifespan while the WSN fulfils a given “mission”. Even if the results are appealing, the control law is centralized and requires a certain level of computational load. Moreover,

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the solution obtained by this approach presents an important number the switches, by often turning on and off certain sensors.

In the present work, we propose another control approach in order to improve the scalability issues though a decentral-ized scheme that reduces both the computational load and the number of switches, while the network lifespan is not decreased when compared to the MPC approach, leading to an improvement in the network power consumption. This control makes use of a Hybrid Dynamical System (HDS) approach [8], where solutions may continuously flow according to some differential equations and may discontinuously jump according to some rules. Therefore, an innovative strategy that fulfils the control objectives is proposed for the control of the functioning modes of the nodes.

The work is organized as follows. Section II is first ded-icated to the system description and, then, we formulate the problem. We present a hybrid dynamic scheduling law for a specific simplified context in Section III, extending it to a general case in Section IV. In Section V, we evaluate in simulation the HDS control strategy on a realistic benchmark and we compare it with an MPC approach also provided [9]. In Section VI, we finally validate the control strategy in experiments through a test-bench. Section VII summarizes the main results and proposes future work directions.

Notations: Throughout the work, N∗ denotes the set of positive natural numbers. Vector eh denotes column h of the

identity matrix, or the hth vector of the Euclidean basis.

II. WSNSYSTEM MODELLING

A. System description

Consider a WSN that contains n ∈ N∗ sensor nodes Si,

i = 1, . . . , n, powered by batteries. The nodes may also be equipped with a harvesting system. All nodes are functionally equivalent; thus they are interchangeable but their hardware can differ, e.g. batteries, processors may be unalike. The communication can be multi-hop or single-hop clustered (see [10]): each node sends its data through a gateway to the supervisor. The nodes can exhibit different functioning modes Mh, h = 1, . . . , m, m ∈ N∗, characterized by a known

power consumption over a given period of time. Typically, the functioning modes are “Active”, “Standby”, etc. This system model includes an Unreachable condition, i.e. during the lifespan of the WSN, some nodes Simay become unavailable.

This situation can occur because of node physical damages, a lack of power resources or strong perturbations of the radio channel. The sensor nodes can exit from this mode when, for instance, the battery is recharged by the harvesting system, the physical damages are repaired or the radio channel is back to normal condition. Due to this unpredictable appear-ance/disappearance of nodes, it is mandatory to supervise the number of reachable Siat every instant time in order to collect

enough measurements from the application viewpoint at the supervisor side. The supervisor chooses the functioning mode of each node thanks to an energy management control strategy presented hereafter.

In this wireless sensor network context, consider that each node Siis characterized by two states xiand ui, and an output

yi:

• xi(t, j) ∈ R>0 is the remaining energy in the node

battery, where t is the continuous time and j is the total number of jumps of the solution, with the constraint on the state xi:

0 < xi6 xi(t, j) 6 xi, (1)

where xi and xi are the lower and upper bounds of the

battery capacity. xi(0, 0) (i.e. the state value at t = 0 and

j = 0) denotes the initial remaining energy.

• ui = ui1 · · · uih · · · uim T

∈ {0, 1}m denotes

the control states, related to the functioning mode of the node. The components of ui are equal to 0 or 1.

Thus, ui = eh means that Si is in mode Mh. Moreover,

ui= 0m denotes the Unreachable condition of node Si

and ui(0, 0) denotes the initial control state. We ensure

that the values of ui(0, 0) are chosen so as to take into

account the “mission” (described hereafter) and possibly to penalize the sensor nodes with smaller lifespan.

• yiis the measurement of the remaining energy in the node

battery delivered by each node at each sampling time kTc

to the supervisor. The supervisor needs this measurement to calculate the control. Therefore, at time kTc:

xi6 yi 6 xi. (2)

Moreover the node Si is characterized by:

• an exogenous input αi ∈ {0, 1} that denotes the

Un-reachable condition for the state xi. αi is 0 (resp. 1) if

Si is able (resp. Reachable). This Unreachable condition

can occur because of node physical damages, strong perturbations of the radio channel or when xi ≤ xi

among others. Note that when xi= xi= yi, the lifespan

of node Si (see (3) below) is equal to zero. In this

situation we consider that the node is Unreachable.

• a power consumption (line) vector Bi ∈ Rm>0. The

component bihof Bidenotes the power consumed by Si

when operating in mode Mh, that is when ui= eh. From

a practical viewpoint, the components of Biare assumed

unequal which is consistent with what is observed for current commercial nodes;

• a disturbance input wi ∈ {0; 1} that corresponds to the

ability for the node Sito harvest energy. This disturbance

input cannot be controlled but may be predicted in some situation. wi = 1 (resp. wi = 0) is associated to the

capability for the harvesting system to harvest (resp. not to harvest) energy from the environment;

• the harvested power value Ei corresponds to the amount

of power provided by the harvesting system of node Si.

Note that Ei is in essence a time-varying value in

real-life conditions. Ei may be in some situations predicted

or even measured;

• the switching power consumption δih→r ∈ R>0 between

two functioning modes takes into account the fact that switching node Si from mode Mh to mode Ml has an

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The lifespan of the Reachable sensor node Si at time t can

be computed by the supervisor as: Li =

yi− xi

Biui

. (3)

B. Control objectives

A dynamic energy saving control policy has to be imple-mented at the application level in order to increase the WSN lifespan while meeting a given “service” requested by the application built on top of the WSN and that makes use of the measurements provided by the nodes. For instance, this service is expressed as a given (minimum) number of measurements that must be provided by the WSN over a defined time lapse. Hereafter, this service is called “mission”. Typically, the “mission” has to be guaranteed over a geographically limited area, where each node can exchange its role with another one without decreasing the performance of the whole network.

The control objectives are to extend the lifespan of the WSN by reducing the overall energy consumption of the nodes via an appropriate management of the functioning mode of each node while providing a given “mission”. During the continuous-time evolution of the solution (xi, ui) for each node Si, we must

have the “mission” defined as:

n

X

i=1

αiuTie1= d1, (4)

where d1 ∈ N∗ defines the exact number of nodes in mode

M1 (Active mode) that can be seen as an external reference.

The constraints that define “mission” (4) can be dynamically changed, depending for instance on the time schedule or on external events. This dynamic mission allows to adjust the needs of the application during the system evolution. For instance, during the day period, when people are present in an office (controlled by the proposed strategy), the “mission” can by defined as d1= a < n nodes in the Active mode while

during the night period, when there is nobody in the office, the “mission” could be d1= b < a.

C. Hybrid representation and pairwise jump rules

We start our modeling framework in a context where no failures are considered (scalars αi are disregarded) and no

harvesting is in place (inputs wiare all zeros). This simplifies

our initial analysis.

Within the above setting, the flow dynamics of the n nodes are given by:

 ˙ xi = −Biui, ˙ ui = 0m, i = 1, ..., n, (x, u) ∈ C, (5) where the flow set C is to be designed. The state dynamics x can be compactly written as:

 ˙x = −Bu, ˙u = 0nm, (x, u) ∈ C (6) where x = x1 x2 · · · xn T ∈ Rn >0, u = u1 u2 · · · un T

∈ {0, 1}nm are the states and, B =

diag(B1, B2, · · · , Bn) ∈ Rn×nm.

The jump dynamics of the n nodes comprise the possibility that the available (Reachable) nodes autonomously decide to swap their respective role within the network. Given (5), one readily understands that swapping role means simply swapping the values of ui. Then, we may define the sets Dil to provide

conditions under which two nodes Si and Sl are required to

swap their roles, under the straightforward assumption that Dil = Dli, so that swapping is simultaneously enabled from

both sides.

The adopted paradigm intrinsically defines a distributed scheduling paradigm, as long as one restricts the sets Dil

to be non-empty (therefore relevant for the potential swap evaluation) only for pairs (i, l), belonging to the edges of a suitable undirected interconnection graph G = (N , E ) characterizing the nodes.

In the general case, the following sets will be designed: Dil= Dli, ∀ nodes Si, Sl: (i, l) ∈ E , (7)

where E is the set of all edges in the interconnection graph. Based on the sets in (7), which will be designed in Section III, we can represent the swapping as an instantaneous update of the state of the nodes Si and Sl, corresponding to:

           x+ i x+l  = xi− (u + i )T∆iui xl− (u+l ) T lul  , u+ i u+l  = ul ui  , (x, u) ∈ Dil, (8)

with the switching consumption matrix ∆i defined as:

∆i =    0 δ2→1 i · · · δih→1 · · · δm→1i .. . . .. ... δ1→m i δi2→m · · · δih→m · · · 0   ∈ R m×m. (9) Note that for swapping, the exogenous inputs αi and αl of

nodes i and l are necessary equal to 1.

Equation (8) only indicates the instantaneous swap for a pair of nodes, but for a complete description of the dynamics, we should also specify that across these jumps all other nodes Si,

i = 1, ..., n − 2 do not experience any change of their xi and

ui states. In particular, the relations in (8) may be compactly

represented as: x+ u+  =g il x(x, u) guil(x, u)  =: gil(x, u), (x, u) ∈ Dil, (10) where gil : (R × {0, 1}m)n → (R × {0, 1}m)n can be

straightforwardly expressed from (8). With the pairwise rules in (10) associated to a jump set Dsw corresponding to the set

where at least one pair of nodes is ready for a swap operation (namely x ∈ Dsw if x ∈ Dil for at least one pair i, l):

Dsw= [ (i,l)∈E Dil, Gsw(x, u) = [ (i,l)∈E: (x,u)∈Dil gil(x, u), (11) where, by construction, Gswis a set-valued mapping (multiple

pairs may be ready to swap at the same time) that possesses the useful property of having a closed graph because its graph is the union of the (closed) graphs of gil.

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Now, we focus on the solutions to (5) and (11), insisting that they evolve in a specific set O where the remaining energy in the batteries of each node is positive for all nodes, and the input vector u has components equal to 0 or 1. Within this set, the flow set C is the closed complement of the flow set D = Dsw relative to O. More specifically:

O = Rn

>0× {0, 1}

nm, C =O \ D \O. (12)

Within the set O, due to the positivity of the entries in the vectors Bi and the constraints on the state values xi, it is

evident that all solutions will be bounded and not complete (their domain is bounded). Thus, the objective is to design the jump sets Dil in an intuitive way, the goal being to

maximize the length of the solution domain in the ordinary time direction, i.e. the WSN lifespan, until that the “mission” (Eq. (4)) is no more satisfied. The WSN lifespan is named hereafter the lifespan of the solution.

III. SCHEDULING LAW FOR THE FAULT-FREE CASE WITH TWO NODES

In the previous section, the WSN power management has been expressed as a control problem that decides the func-tioning mode for each node, with the design of the pairwise sets Dil. Indeed, dynamics (5), (10) already clarify what is

happening during the flowing (i.e. when the batteries are discharging) and what should happen at each reconfiguration of the network. Basically, when the state (x, u) belongs to Dil, the nodes Si and Slswap their roles. To suitably design

Dil for all i 6= l, the adopted control paradigm focuses on a

pairwise reconfiguration rule.

This rule is described for the simplified case of two nodes S1, S2without harvesting systems (w(t) = 0n) and two modes

M1, M2 (resp. Active and Standby), with D12 = D21 = D.

For this specific case, an optimal result is proved hereafter. Moreover, in this case, it does not make sense to have an Unreachable state because no replacement node is available. Therefore, we focus on the fault-free two-nodes case, with αi= 1 ∀ i = 1, 2. Moreover, d1= 1.

The rationale behind the choice of D12 is that we would

like to design an algebraic condition on x and u that expresses when it is convenient to swap roles between both nodes in such a way to maximize the lifespan of the solution. This quantity may be expressed as a cost function J to be maximized. When designing D12, the expected lifespan if the solution does not

perform further jumps:

J (x, u) = min

i:ui6=0m

xi− xi

Biui

(13) is introduced. A first condition to encode in the flow set is that it is not convenient to jump (or swap roles) whenever J (x+, u+) < J (x, u). Thus, one must ensure that:

J+(x, u) := J (gx(x, u), gu(x, u)) < J (x, u)

⇒ (x, u) /∈ D12.

(14) Intuitively, the condition (14) means that no switch is per-formed if it reduces the lifespan. Note that this condition is a function of both x and u.

Even though condition in (14) is reasonable, it may still induce undesirable behaviours in some cases. For instance, consider two identical nodes, namely ∆1 = ∆2, B1 = B2.

The Active (resp. Standby) mode is supposed associated with a large (resp. small) power consumption. Assume also that ∆i

are relatively small when compared to the power consumption of the Active mode. In this case, the best strategy is clearly to keep one node in the Active mode until its battery is drained, and then swap once and only once along the solution. However, picking D12 as the closed complement of the left

hand condition of (14) will enforce extra unnecessary jumps as soon as the energy level in the active node becomes sufficiently small compared to the energy in the other node, and vice-versa, and so on.

One way to avoid this situation is to encode in D12another

condition: nodes will not swap if there is no “emergency” to do so. Indeed, when the left hand condition in (14) holds, solutions will still keep flowing (thus, no switch is applied) if waiting any further does not cause any reduction in the expected lifespan after the potential switch. To characterize this reduction, denote by i∗ the index (or the set of indexes) that minimizes J+, namely:

i∗= arg min i=1,2 x+i Biu+i = arg min i:ui6=0m xi− uT3−i∆iui Biu3−i . (15) Note that i∗ may contain both indexes if the two expected lifespans coincide. Then, characterize the reduction as the derivative of J+during flow (if no jump would be performed):

δJ (x, u) = min l∈i∗ ˙ x+l Blu+l = min l∈i∗ −Blul Blu3−l . (16)

Note that the derivative of uT3−i∆iui is zero along the flow.

The function in (16) captures the idea of how damageable it is to postpone the swap. In other words, (16) expresses how much smaller J+ will become if the solution flows for some

extra time, before jumping. Since the two components in Bi

are supposed not equal, then δJ (x, u) 6= −1.

Clearly, if we keep flowing, the decrease rate of J will simply be 1 as the lifespan decreases linearly as time flows. Therefore, one extra criterion for the selection of C may be:

δJ (x, u) ≥ −1 ⇒ (x, u) /∈ D12, (17)

which is well defined, as indicated above, because δJ (x, u) 6= −1. In particular, what happens along solutions, as long as J+≥ J is that δJ(x, u) > −1 whenever there is no urge to jump, and then once the “argmin” in (15) changes, we start getting δJ (x, u) ≤ −1 and a jump occurs.

To summarize, one selects:

D12= {(x, u) : J+(x, u) ≥ J (x, u) and δJ (x, u) ≤ −1}.

(18) Note that the definition is commutative that is, there is not specific role of nodes S1 and S2 in the selection of D12. The

selection (18) enables to prove the following optimality result. Notice that hereafter xi= 0 ≤ xi ≤ xi. If this is not the case,

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Theorem 1. Assume that Bi,i = 1, 2 has positive and distinct

components and that matrices ∆i, i = 1, 2 have strictly

positive off diagonal elements. Consider any initial condition with positive values of xi(0, 0) and with ui(0, 0), i = 1, 2

being two independent columns of the identity matrix. The solution of (5), (11), (18) is unique and it has a lifespan equal to the maximum lifespan that can be obtained by selecting arbitrary jump times for(5), (8) starting from the same initial conditions.

Proof. The proof is carried out by first (step 1) showing that the (optimal) solution providing the maximum lifespan performs at most one jump (one swap between both nodes), and then (step 2) showing that the solution generated by the proposed jump rule is optimal.

Step 1. Assume that the optimal solution ϕopt =

(xopt, uopt), having lifespan T , performs more than one jump

(or swap) and denote by t1 and t2 the first 2 jump times.

These jump times clearly satisfy 0 ≤ t1≤ t2≤ t3, where we

denote by t3 either the next jump time, or the lifespan time

t3 = T . Since T is the lifespan of the solution, we must

have xopt,1(T, j) > 0, xopt,2(T, j) > 0, and that at least

one of them is zero, where j is the total number of jumps of the solution. Consider now another solution (x∗opt, u∗opt)

starting from the same initial condition (x(0, 0), u(0, 0)) = (x10, x20, u10, u20) and performing one jump at time t∗1 =

t1+ (t3− t2), and then performing the same jumps as the

original solution does, namely t∗2 = t3, . . ., t∗j−1 = tj. Note

that this new solution performs one more jump less than the original one. Then, it can be proved that at ordinary time t∗2 = t3 (and before the possible corresponding jump), both

outputs x1 and x2 of the “*” solution are strictly larger than

the corresponding states of the original opt solution. As a consequence:

0 < xi(T, j) < x∗i(T, j − 1), i = 1, 2, (19)

which implies that the lifespan of the “*” solution is larger, thus completing the proof of this step. We only prove (19) for the case i = 1 as the case i = 2 is identical. To this end, consider: x∗opt,1(t∗2, 1) = x10− (t∗1− 0)B1u10− (t∗2− t ∗ 1)B1Πu10− (Πu10)T∆1u10= x10− (t1+ (t3− t2))B1u10− (t2− t1)B1Πu10− (Πu10)T∆1u10,

where the swaping matrix Π := [0 1

1 0] and t∗2− t∗1= t2− t1 is

chosen. Instead, for the original solution we have:

xopt,1(t3, 2) = x10− (t1− 0)B1u10− (t2− t1)B1Πu10−

(t3− t2)B1Π2u10− (Πu10)T∆1u10−

uT10∆1Πu10= x10− (t1+ (t3− t2))B1u10−

(t2− t1)B1Πu10− (Πu10)T∆1u10− uT10∆1Πu10,

with Π2 = I. This clearly shows that the second state

is smaller because it performs one extra jump and the off diagonal terms of ∆i, i = 1, 2 are all positive.

Step 2. Consider now the structure of the jump set in (18) and note that all along any solution ϕ(t, j) = (x(t, j), u(t, j)),

the function ψ(t, j) := t + J (ϕ(t, j)) denotes the envisioned lifespan (in the future) of the solution in the case when no other jump happens, while ψ+(t, j) := t + J+(ϕ(t, j)) denotes the

envisioned lifespan of the solution if a single jump happens at the current time. This is easily understandable from the fact that J and J+ measure the remaining lifespan, so the

total amount of the envisioned future flow. Then, at each hybrid time instant (t, j) in dom ϕ (hybrid time domain [8]), the envisioned lifespan corresponds to such an envisioned remaining lifespan plus the already elapsed time t. As an additional property, note that along any solution, the function ψ is non-decreasing. Indeed, its derivative along the flows is trivially zero, while the first condition in (18) implies that it is non-decreasing at jumps. Moreover, it is straightforward to conclude from (16) that:

d dtψ

+(t, j) = 1 + δJ (ϕ(t, j)),

(20) which can never be zero because of the assumption that the two components in the matrices Bi are not equal. We now

split the proof in two cases comparing the optimal solution to our solution and establishing that they have the same lifespan. Case 1: If the optimal solution ϕopt performs no jumps, then

the envisioned lifespan from the initial condition is already T , namely ψ(0, 0) = J (ϕ(0, 0)) = J (ϕopt(0, 0)) = T . As

a consequence, our solution will flow for at least (therefore exactly, from optimality) T ordinary time because the function ψ is non-decreasing.

Case 2: If the optimal solution ϕoptperforms one jump, denote

by t∗ ∈ [0, T ) the time of that jump and note that it must satisfy:

t∗+ J+(ϕopt(t∗, 0)) = t∗+ J (ϕopt(t∗, 1)) = T. (21)

Denote also ψopt(t, j) = t + J (ϕopt(t, j)) . Since the solution

flows for all (t, j) ∈ [0, t∗) × {0}, and since dtdψ+optcan never

be zero as emphasized after (20), it must necessarily be that

d dtψ

+

opt(t, 0) > 0 for all t ∈ [0, t∗). Otherwise there would

be another solution jumping before t∗ and lasting longer than ϕoptwhich is impossible due to optimality. As a consequence,

since our solution ϕ starts from the same initial condition as ϕopt, it must hold, also based on (20), that 1 + δJ (ϕ(0, 0)) =

d dtψ

+(0, 0) = d dtψ

+

opt(0, 0) > 0, meaning, according to the

second condition in (18), that our solution does not jump at the initial time. Due to the uniqueness of solution of our flow dynamics, we also conclude that 1+δJ (ϕ(t, 0)) > 0 for all t ∈ [0, t∗], thus implying that our solution behaves optimally until t∗. The interval is now closed due to continuity of the solution along the flows. Then, we have from (21) that ψ+(t∗, 0) = t∗ + J+(ϕ(t∗, 0)) = T , which completes the proof by the non-decreasing property of ψ established above.

IV. SCHEDULING LAW FOR THE GENERAL CASE

The approach of Section III is now extended to the case with n > 2 nodes, m ≥ 2 modes, the Unreachable condition and harvesting systems.

In a real-life condition, the WSN system control is not based on the state xi but on the measurement of outputs

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the harvesting action we now take into account the inputs wi i = 1, ..., n quantifying the harvested energy at node i.

In particular, we generalize (5) to    ˙ xi = −Biui+ Eiwi, yi = αixi, ˙ ui = 0m, i = 1, ..., n, (y, u) ∈ C, (22) which can be compactly written as

   ˙x = −Bu + Ew, y = αx, ˙u = 0nm, (y, u) ∈ C (23) where w = w1 w2 · · · wn T ∈ {0, 1}n, y = y1 y2 · · · yn T ∈ Rn >0, α = diag(α1, α2, ..., αn) ∈ {0, 1}n×n, and E = diag(E 1, E2, · · · , En) ∈ Rn×n. Note that

the Unreachable condition for node Sicorresponds to αi= 0,

ui= 0mand wi= 0. Likewise, jump rules now only depend

on the accessible measurements y. Within this context, Jil,

Jil+, kil∗, δJil can be defined as those in (13), (14), (15), and

(16), respectively, for each (i, l) ∈ E as follows: Jil(y, u) := min

k=i,l; k:uk6=0m

yk− Xkmin

Bkuk

(24) Jil+(y, u) := Jil(gily(y, u), g

il u(y, u)) (25) kil∗ = argmin k=i,l; k:uk6=0m yk− uTkc∆kuk Bkukc (26) δJil(y, u) = min k∈k∗ il −Bkuk Bkukc , (27)

where kcrepresents the “second” node in a pair, namely kc= i if k = l, and vice-versa.

With the above definitions, the pairwise jump sets Dil can

now be defined by suitably generalizing the expression in (18): Dil= {(y, u) : Jil+(y, u) ≥ Jil(y, u)

and δJ (y, u) ≤ −1 and αi= αl= 1},

(28) where the swap is now inhibited if any of the two nodes in a pair is in the Unreachable condition. Selection (28) is then completed by the definition of Gsw and Dsw in (11).

Jump rules must now be defined to take into account the presence of the exogenous input α, that captures the possibility for nodes to fall into the Unreachable condition, and the dynamically changing “mission” during the system evolution. Thus, for each node, three supplementary jump rules are defined.

The first one corresponds to the situation when the node Si is in a mode Mh and the corresponding αi goes to zero,

i.e. the node becomes Unreachable). Then, the input ui is

automatically switched to the Unreachable value, i.e. ui= 0m:

 x+i = xi u+i = 0m, (y, u) ∈ D 0 i, (29a) D0i := {(y, u) : ui= eh and yi≤ xi}, (29b) D0:= [ i=1,...,n D0 i, G0(y, u) = [ i: (y,u)∈D0 i gi0(y, u). (29c)

In this case, note that the supervisor did not receive informa-tion related to the remaining energy xi for the Unreachable

node with αi = 0. Moreover, g0i(y, u) has a similar

struc-ture to functions gil(x, u) introduced in (10). In particular,

g0

i(y, u) : (R × {0, 1}m)n→ (R × {0}m)n is a compact way

of representing the jump map in (29a) essentially comprising identities in all entries except for those related to node Si.

The second jump rule corresponds to a more sophisticated action ensuring that the “mission” is fulfilled even when an Active node becomes Unreachable, and in the presence of nodes in the Standby mode. This means that there are nodes in mode Mh where h 6= 1. This second jump rule forces one

Reachablenode (without any pre-specified priority) to become Activeif the “mission” is no more satisfied. This will typically happen if an Active node falls in the Unreachable condition because of a jump arising from (29). This jump rule is given by:  x+i = xi u+i = e1, (y, u) ∈ D1i, (30a) D1 i := {(y, u) : ui= eh6=1 and n X i=1 αiuTie1≤ d1− 1}, (30b) D1:= [ i=1,...,n D1i, G1(y, u) = [ i: (y,u)∈D1 i g1i(y, u), (30c) where gi1(y, u) : (R × {0, 1}m)n → (R × {0, 1}m)n is a

compact way of representing the jump map in (30a) essentially comprising identities in all entries except for those related to node i. Basically, g1

i(y, u) has a similar structure to functions

gil introduced in (10).

The third jump rule corresponds to the situation when the “mission” changes dynamically, i.e. d1 is a time-varying

integer function. In this case, node Si jumps to the Standby

mode from the Active one:  x+i = xi u+i ∈ {eh: h 6= 1}, (y, u) ∈ D2i, (31a) D2 i := {(y, u) : ui= e1 and n X i=1 uTie1≥ d1+ 1}, (31b) D2:= [ i=1,...,n D2 i, G2(y, u) = [ i: (y,u)∈D2 i g2i(y, u), (31c) where g2 i(y, u) : (R × {0}m)n → (R × {0, 1}m)n is a

compact way of representing the jump map in (31a) essentially comprising identities in all entries except for those related to node i. It has a very similar structure to function gilintroduced

in (10). In the case, where d1(t1) < d1(t2) with t1< t2, the

jump rule (30) is executed.

For these jump rules (11), (28), (29)-(31), the energy in the battery of node Sidoes not experience any discontinuity, thus

x+i = xi, where xi ∈ [xi, xi]. Remember that if yi = xi,

the node Si has the lifespan Li = 0 at the time t (see (3)),

therefore it is considered as an Unreachable node.

With the definitions in (11), (28), (29c), (30c), (31c), the complete jump dynamics can be compactly represented as:

x+

u+



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where D = Dsw∪ D0∪ D1∪ D2, and G is a set-valued map

corresponding to the outer semicontinuous hull of Gsw, G0,

G1, G2 (the outer semicontinuous hull is a map whose graph

is the union of the graphs of the composing maps) namely allowing for a jump whenever one of the corresponding rules is enabled. The discrete-time dynamics (32) together with (22) correspond to a hybrid description of the WSN system under the specific control action.

We may now express the following result, which is a straightforward consequence of the jump rules in (29) - (31) and of the fact that the jumps from (28) have no effect on the “mission” because they merely correspond to swapping the states of nodes Si and Sl.

Proposition 1. Given any solution to (5), (32), if at least d1

nodes are not in theUnreachable condition at each time in its domain, then the “mission” (4) is always satisfied during the flows of the solution.

The proof of the proposition is omitted because it is a straightforward consequence of the definition of jumps rules and of the fact that flowing is forbidden unless the “mission” is accomplished.

V. SIMULATION OF THEHDSSTRATEGY

To validate the theoretical results presented above, some simulations on a realistic benchmark are performed to illustrate the features of our control law based on HDS. The proposed scenario is built from device data-sheets and laboratory mea-surements. Firstly, we will describe the simulation conditions. Then, we will discuss the results provided by our HDS strategy and, they will be compared to another strategy based on MPC previously proposed by the authors [6].

A. Simulation conditions

We validate our proposed energy management strategy with a WSN composed of n = 6 nodes that may possibly embed a harvesting system. Each node can stay in one of three states, namely, m = 2 functioning modes and the Unreachable condition. The two functioning modes are defined by the real node abilities. This means that they can be placed in the Active or Standby mode, M1 and M2 respectively. Moreover, they

may fall in the Unreachable condition due to environmental disturbances as communication breakdown, or an insufficient energy level. This context is similar to the one presented in [7].

The “mission” is defined by d1= 3 nodes in Active mode

(M1), which is considered sufficient to guarantee the service

that the WSN must provide in its given application.

The components of matrix Biin (5) are computed from the

information provided in Table I. Likewise, Table II provides the initial energy capacity of the batteries associated with each sensor node and, the harvesting system availability.

B. Simulation of the HDS strategy

The simulations of the WSN with the proposed HDS control strategy are performed in the MATLAB/Simulink environment and the HyEQ Toolbox [11].

Table I: Average energy consumption Bi, [mAV · hour] of

node Si in the functioning modes M1 and M2.

Sensor node Mode M1 Mode M2

S1 34,854 5,846 S2 36,482 6,031 S3 36,593 6,105 S4 36,482 6,105 S5 36,556 6,105 S6 33,041 5,735

Table II: Node battery and harvesting system characteristics

Node Nom. battery capacity, [mAV ·hour] Harvesting availability Ei, [mAV · hour] Energy coef. ξi, [1] Harvesting period, per 24 hours S1 3885 – 1 – S2 3885 – 0,8 – S3 3885 77,7 0,9 7-12 S4 3515 – 0,7 – S5 3515 99,9 1 13-18 S6 3515 – 1 –

Figure 1 shows the evolution of the functioning mode for each node when the HDS control is applied. Note that nodes S3 and S5 present harvesting capability, which explains the

oscillatory evolution of their remaining energy, depicted in Figure 2. We can observe that the remaining energy of all nodes is never equal to 0. Basically, this is related to the constraints (1) on the state xi that avoid to fully drain the

battery. From both Figures 1 and 2, we see that the “mission” (d1 = 3 nodes in mode M1) is fulfilled approximately until

196 hours, which represents the WSN lifespan.

During the simulation, the nodes jump between mode M1

and mode M2, following the rule (22)–(28). When a sensor

node has no more enough energy, it falls in the Unreachable condition, following the rule (29). This same node may jump back to the Reachable condition at any time and be taken into account by the rule (28), which is decentralized.

Figure 1: Evolution of the functioning modes of the nodes.

C. Comparison among HDS and other strategies

Now, we will compare the proposed HDS approach with other strategies, namely, a “basic controller” and a MPC-based one. Both of them are discussed in [6]. The “basic

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Figure 2: Evolution of the remaining energy in each node battery.

controller” manages the node activity in a centralised way under the constraint of fulfilling the “mission” and without any other constraint. On the other hand, the MPC considers all constraints mentioned in this work. However, small differ-ences between the HDS approach and the MPC one must be highlighted (see [6] for details), namely, the MPC:

• is a centralized controller that penalizes the most con-suming nodes;

• the switching energy loss (δik→l) is integrated in bil [6].

For comparison purpose, we consider the WSN benchmark without and with the harvesting profile described above. Likewise, we will use two MPC formulations, namely a Mixed Integer Quadratic Programming (MIQP) and a Mixed Integer Linear Programming (MILP), this latter having some computational advantages due to the particular structure of the problem to be solved.

A summary of these strategies is shown in Table III (see [12] for details). It must be highlighted that the results obtained with the control strategies based on MPC and HDS approaches are promising. Compared to the “basic controller’, the WSN lifespan obtained:

• using the MPC approach is 37% longer without harvest-ing and 49% longer with harvestharvest-ing;

• likewise, using the HDS approach it is 28% longer without harvesting and 39% longer with harvesting. Moreover, for the specific benchmark case, the WSN lifespan obtained using the MPC approach can be longer by 4.1% and 8.5% without and with (resp.) harvesting systems compared to the HDS approach. This phenomenon can be explained by the characteristics of the different controllers. On the one hand, the strategy based on MPC does not limit the number of switches compared to the strategy based on HDS, where a switch is enforced only if it significantly extends the WSN lifespan. As a consequence, we get a larger number of switches by using MPC, as shown in [6]. The reduction of the number of switches allows to further reduce the energy consumption (especially the one due to the switches) and to not increase the overload of the radio channel, because the number of communications also decreases. In fact, if we need to swap roles between nodes, extra message must be sent between the supervisor and the nodes.

Table III: Comparison of scenarios in terms of WSN lifespan and number of switches.

Strategies WSN lifespan without harv. sys., [hours]

WSN lifespan with harv. sys., [hours] Number of switches Basic 128 192 3 MPC/MIQP 175 287 ≈ 102 MPC/MILP 171 284 ≈ 102 HDS 164 266 7 (5)

We can see that the benchmark presents different levels of complexity depending on the number of nodes. Therefore, we have an impact on the computational workload according to the implemented control strategy. Figure 3 shows the simula-tion time to solve the feedback algorithm for a control period Tc= 1 hour (one control step) for different numbers of sensor

nodes. Note that the simulated system with the MPC strategy is discrete and, thus, a difference equation is computed. However, the simulated system with the control strategy based on HDS is hybrid (continuous/discrete). Thus, differential equations must be numerically integrated and, the control algorithm must be discretized (here we select a sampling time Tc= 1 hour). The

computational times are obtained in the Matlab environment using the tic-toc function. Matlab runs on one core of the Intel Xeon Processor E5620 with 12 MB of cache, 2.40 GHz and 5.86 GT/s [13]. From these computational times, even if the evaluation is rough, it is evident that the control strategy solved with MIQP is more “demanding” compared to the problem solved with MILP. The comparison between HDS and MPC approaches shows that the HDS algorithm is in the range of two orders of magnitude faster than the MIQP approach for the benchmark with n = 6 nodes.

Figure 3: Complexity analysis of the algorithms.

To summarize, the advantages of using our developed HDS approach are:

• the control strategy is scalable and reliable;

• the number of switches is reduced and,

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VI. EXPERIMENTAL RESULTS

The energy management strategy based on HDS described in this paper is now implemented in a real test-bench. This strategy consists of continuous physical processes of the energy charge/discharge of the node batteries (described by xi) and, of finite-states for the functioning modes (described

by ui). Note that in real-life conditions, the remaining energy

in the node battery is supposed to be measured and delivered by each node at each sampling time, i.e. we have yi = xi at

each time kTc. In the present experiment, Tc is chosen equal

to 1 hour.

A. Test-bench description

The hardware test-bench considered here consists of one supervisor, one router and n = 6 sensor nodes. The supervisor is a laptop where the control strategy is implemented. It is equipped for communication with a WiFi card. The WiFi router is used to increase the range of the network and to form the infrastructure topology. The sensor nodes are Flyport WiFi 802.11g modules developed by OpenPicus [14] connected to a sensing element. Each node has a battery, as shown in Figure 4. The sensing element contains a temperature & humidity sensor DHT-11 [15]. The Flyport WiFi 802.11g module is a programmable system-on-module with integrated WiFi 802.11g connectivity. For a programmed sensor node, two functioning modes are considered, namely the Active mode M1 and the Standby mode M2 that are described as

follows:

• in the Active mode M1, sensing, computing and

com-munication units are “duty cycled” [16] with a sampling period Ts = 1min to sense, process and exchange data

with the supervisor;

• the Standby mode M2is similar to mode M1, i.e. it is duty

cycled, but with a much larger sampling period Tw 

Ts. In this mode, the node remains in the sleep state

and, it wakes up each Tw = Tc = 1 hour to receive the

commands from the supervisor and monitor the remaining battery capacity.

Based on the technical data-sheet of the Flyport module and on laboratory measurements, the numerical values of the energy consumption of the nodes were estimated. They are presented in Table I.

Figure 4: Used sensor nodes.

Each node presents one type over two Li-polymer recharge-able batteries [17], namely, type 1 for sensor nodes S1− S3,

and type 2 for S4 − S6. Note that harvesting systems are

not available. The Li-ion batteries have nominal capacities xi = 3885mWh for type 1, and xi = 3515mWh for type 2.

These batteries embed an electronic protection circuit ensuring a minimum State-of-Charge (SoC) value corresponding to 10% of the nominal capacity for type 1 batteries, and corresponding to 16% of the nominal capacity for type 2 batteries. As a consequence, the lower energy limit of the sensor nodes using a type 1 battery is equal to xi= 3885mWh · 0.1 = 388, 5mWh (for nodes i = 1, 2, 3). Instead for a type 2 battery, it is xi= 3515mWh · 0.16 = 562.4mWh (for nodes i = 4, 5, 6).

After the battery calibration step described in [18], an accurate experimental model of the battery V oltage − SOC curve is built. Figure 5 depicts an example of such SoC profiles for both types of new batteries, at 23◦C, which corresponds to the typical environment temperature in the office where the sensor nodes are deployed. This calibration step together with the protection circuit allows to safely (i.e. without damaging the battery) and efficiently exploit the battery capabilities. The estimation of the remaining energy in the battery of all nodes is implemented together with the control law.

10 20 30 40 50 60 70 80 90 100 2.5 3 3.5 4 SOC [%] Voltage [V] Type 1 Type2

Figure 5: SOC profiles for both battery types. The coordination between the sensor nodes and the supervi-sor is realized via the router, which uses the LINC coordination environment [19]. LINC is a resource-based middleware with a particular emphasis on the sensor/actuator network field. The LINC middleware addresses issues raised by applications considering a large number of sensor nodes (i.e. up to several hundreds), distributed over a wide field (e.g. a building or a set of buildings) and, connected via heterogeneous and unreliable communication protocols (e.g. various wireless networks) [20]. B. Control objectives

For this test-bench, the WSN with 6 sensor nodes has been deployed in a working office as shown in Figure 6. In order to control the air conditioning system, temperature and humidity are sensed through the sensor nodes. During the day, when the office is in use, the control of the air conditioning system requires measurements from 3 sensor nodes. During the night, measurements from only one sensor node are enough to ensure the appropriate control of the air conditioning unit. Therefore, the “mission” is split in two phases corresponding respectively to “day” and “night” periods of time. Then, the constraints that define the “mission” have to be dynamically adjusted, depending on the time schedule, leading to a dynamic “mission”, as summarized in Table IV.

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Table IV: Definition of the dynamic mission

Time period d1 Objectives

Day 8am−6pm 3 3 nodes in mode M1

Night 6pm−8am 1 1 node in mode M1

Figure 6: WSN deployed in a working office.

At time instant t = 0, all the sensor nodes of the system are in the Active (M1) mode. Basically, they must transmit the

initial remaining energy in their battery. Then, the supervisor checks whether the node batteries have enough energy so that any node Sican fulfil the “mission” (i.e. being placed in mode

M1). If this is true, during the day period, 3 nodes are placed

in mode M1while the other ones that are Reachable are placed

in M2. During the night period, only 1 node is placed in mode

M1 and the others are placed in mode M2.

When a sensor node Si is not seen by the supervisor, it

is considered in the Unreachable condition. This situation occurs for instance when the remaining energy of a sensor node is lower than or equal to xi, or because of any other faulty conditions. Then, the control law assigns new modes to the remaining nodes in order to meet the dynamic “mission” while reducing the energy consumption of the sensor network. Note that this is the only centralized decision in our control law while all the other decisions are distributed. When the supervisor receives again information from a node that was beforehand in the Unreachable condition, it places this node in mode M1 or in mode M2 depending on the “mission”

fulfilling.

C. Experimental results

The energy management strategy based on the HDS ap-proach is now tested on the described test-bench. For this purpose, the control strategy is also written in Python and integrated in the LINC middleware.

The switching consumption matrices are supposed equal for all nodes. They are given by:

∆i=  0 0.01 0.05 0  mJ. (33)

Note that the energy consumed for a switch is very hard to measure. It depends on the sensor node itself and on the environment conditions, especially the communication disturbances. However, the selection of ∆iin (33) corresponds

to an indirect way to penalize switches.

0 10 20 30 40 50 S6 S5 S4 S3 S2 S1 time [h] Mode 1 − Active Mode 2 − Standby Unreachable condition

Night Day Night Day

Figure 7: Evolution of the functioning mode of the nodes.

Figure 8: Estimated remaining energy of each sensor node.

The implementation in real-life conditions starts at 5 p.m. and lasts 50hours. The evolution of the functioning modes and of the estimated remaining energy of all nodes are presented in Figure 7 and Figure 8 respectively. It can be observed that the WSN lifespan is equal to 46 hours, where the dynamic “mission” is fulfilled periodically for day and night periods of time. During the WSN lifespan, sensor nodes S4 and

S6 fall in the Unreachable condition, which may be caused

by perturbations of the radio channel. At the end of the experiment, the nodes still present remaining energy in their batteries, as shown in Table V.

Table V: Remaining energy at the end of the experiment.

Sensor node ximWh xi(tfinal) mWh

S1 388,5 388,5 S2 388,5 427,3 S3 388,5 466,2 S4 462,4 562,4 S5 462,4 632,7 S6 462,4 562,4

We can see in Figure 8 that the slope of the curves of the estimated remaining energy is also proportional to the

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energy consumption in the different functioning modes of the nodes. The number of switches is larger in real-life conditions compared to the simulation in Matlab. In fact, this can be the reflection of larger radio channel perturbations and of the dynamic “mission” that imposes extra switches. Therefore, we emphasize the importance of reducing the number of switching with the controller, because in real-life, additional switches may be expected.

VII. CONCLUSION

Wireless Sensor Networks (WSNs) have undergone in the last few years a tremendous growth, both in industry and in academia. This is mainly due to the different potentials of this technology, such as the essential absence of wiring costs and the possibility of application domains that were inaccessible for wired sensor nodes. However, WSNs also need to face significant design challenges because of their limited computational and storage capabilities, and the limited available energy as sensor nodes are usually supplied by a battery. The energy is a critical resource and it often constitutes a major obstacle to the deployment of sensor networks that will be used everywhere in the world of tomorrow.

In this work, a distributed control algorithm for energy management in a WSN has been proposed. The energy in the sensor nodes is modeled using a Hybrid Dynamical System representation. The WSN has to provide a given functionality (named “mission”) while taking into account the possibility for nodes to fall in an Unreachable condition. The optimality of our approach has been proven in the case of two nodes and an extension of the theorem is developed to encompass a more general case considering n nodes with harvesting systems. Simulation results on a realistic benchmark and comparison with an Model Predictive Control approach show the potential of the proposed control strategy, since the present distributed controller reduces the number of switches between two modes and the computational workload, besides making the WSN scalable and reliable. These benefits have been validated in experiments on a real test-bench.

VIII. ACKNOWLEDGMENT

Work supported in part by ANR project LimICoS contract number 12 BS03 005 01, by grant OptHySYS funded by the University of Trento, by the H2020 TOPAs project, nb 676760, and by the ARTEMIS ArrowHead project nb 332987.

REFERENCES

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[3] W. Hailong, S. Yan, and W. Tuming, “Dynamic power management of wireless sensor networks based on grey model,” in 3rd International Conference on Advanced Computer Theory and Engineering, vol. 1, 2010, pp. 133–137.

[4] V. Sharma, U. Mukherji, V. Joseph, and S. Gupta, “Optimal energy man-agement policies for energy harvesting sensor nodes,” IEEE Transactions on Wireless Communications, vol. 9, no. 4, pp. 1326–1336, 2010. [5] G. Anastasi, M. Conti, M. Di Francesco, and A. Passarella, “Energy

conservation in wireless sensor networks: A survey,” Ad hoc networks, vol. 7, no. 3, pp. 537–568, 2009.

[6] O. Mokrenko, M. I. Vergara-Gallego, L. Lombardi, L. Lesecq, D. Pus-chini, and C. Albea, “Design and implementation of a predictive control strategy for power management of a wireless sensor network,” The 14th annual European Control Conference, 2015.

[7] P. S. Sausen, J. R. de Brito Sousa, M. A. Spohn, A. Perkusich, and A. M. N. Lima, “Dynamic power management with scheduled switching modes,” Computer Communications, vol. 31, no. 15, pp. 3625–3637, 2008.

[8] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid Dynamical Systems: modeling, stability, and robustness. Princeton University Press, 2012. [9] O. Mokrenko, S. Lesecq, W. Lombardi, D. Puschini, C. Albea, and

O. Debicki, “Dynamic power management in a wireless sensor network using predictive control,” in 40th Annual Conference of the IEEE Industrial Electronics Society, 2014, pp. 4756–4761.

[10] V. Mhatre and C. Rosenberg, “Homogeneous vs heterogeneous clustered sensor networks: a comparative study,” in IEEE International Conference on Communications, vol. 6, 2004, pp. 3646–3651.

[11] R. Sanfelice, D. Copp, and P. Nanez, “A toolbox for simulation of hybrid systems in matlab/simulink: Hybrid equations (hyeq) toolbox,” in Proceedings of the 16th international conference on Hybrid systems: computation and control, 2013, pp. 101–106.

[12] O. Mokrenko, “Energy management of a wireless sensor network at application level,” Ph.D. dissertation, Universite Toulouse III Paul Sabatier, 2015.

[13] “Intel xeon processor e5620, ark.intel.com,” 2015. [Online]. Available: ark.intel.com

[14] “www.openpicus.com,” 2015. [Online]. Available: http://www.openpicus.com

[15] “www.aosong.com/en/products/details.asp?id=109,” 2015. [Online]. Available: http://www.aosong.com/en/products/details.asp?id=109 [16] S. Du, A. K. Saha, and D. B. Johnson, “Rmac: A routing-enhanced

duty-cycle mac protocol for wireless sensor networks,” in 26th IEEE International Conference on Computer Communications, 2007, pp. 1478–1486.

[17] “www.farnell.com/datasheets/1666650.pdf and 1666648.pdf,” 2015. [Online]. Available: http://www.farnell.com/datasheets/1666650.pdf and 1666648.pdf

[18] O. Mokrenko, M.-I. Vergara-Gallego, W. Lombardi, S. Lesecq, and C. Albea, “WSN power management with battery capacity estimation,” in 13th IEEE International NEW Circuits And Systems conference, 2015. [19] M. Louvel and F. Pacull, “Linc: A compact yet powerful coordination environment,” in Coordination Models and Languages, 2014, pp. 83–98. [20] L.-F. Ducreux, C. Guyon-Gardeux, S. Lesecq, F. Pacull, and S. R. Thior, “Resource-based middleware in the context of heterogeneous building automation systems,” in 38th Annual Conference on IEEE Industrial Electronics Society, 2012, pp. 4847–4852.

Olesia Mokreko was Researcher-Engineer at CEA-LETI in Grenoble, France when this work was per-formed. She received the M.S. degree in Aeronautics in 2013 from the National Technical University of Ukraine Kyiv Polytechnic Institute, and the Ph.D. degree in Automatic Control in 2015 from the University of Toulouse III. Her current research interests include energy management and wireless sensor networks.

Carolina Albea is an associate professor at the University of Toulouse III (Universit Paul Sabatier) from 2011. Her research is performed at the LAAS-CNRS. She received her PhD in automatic control in 2010 from the University of Sevilla, Spain, and the University of Grenoble, France. From 2010 to 2011, she held a post-doctoral position at the CEA-LETI Minatec campus in Grenoble, France, on the control of nanoelectronic circuits. Her research interests cover: nonlinear control, hybrid dynamical systems, consensus, control of electronic devices and control of networks.

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Suzanne Lesecq passed the Agrgation in electrical engineering in 1992. She received the Ph.D. in pro-cess control from the Grenoble Institute of Technol-ogy, in 1997. She joined the University of Grenoble in 1998 where she has been appointed as Associate-Professor from 1998 to 2006 and full-time Associate-Professor from 2006 to 2009. She joined CEA-LETI in mid-2009. Her topics of interest are application of control theory to power consumption management of Many-Processor System-on-Chip and sensor networks, and control of power converters.

Luca Zaccarian has been Assistant and then As-sociate Professor at the University of Roma Tor Vergata (Italy) since 2000. Then in 2011 he be-came Directeur de Recherche at the LAAS-CNRS, Toulouse (France) and since 2013 he also holds a part-time associate professor position at the Univer-sity of Trento, Italy. Luca Zaccarian’s main research interests include analysis and design of nonlinear and hybrid control systems, modeling and control of mechatronic systems. He was a recipient of the 2001 O. Hugo Schuck Best Paper Award given by the American Automatic Control Council and he is a fellow of the IEEE.

Figure

Table II: Node battery and harvesting system characteristics
Figure 2: Evolution of the remaining energy in each node battery.
Figure 4: Used sensor nodes.
Table IV: Definition of the dynamic mission

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For the staggered stud walls the low-frequency sound transmission loss is limited by a mass–air–mass resonance which, as ex- pected, seems to shift lower in frequency as more mass